Optimal. Leaf size=31 \[ \frac{1}{3} \sqrt{2} E\left (\sin ^{-1}(x)|-\frac{3}{2}\right )-\frac{1}{3} \sqrt{2} \text{EllipticF}\left (\sin ^{-1}(x),-\frac{3}{2}\right ) \]
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Rubi [A] time = 0.0296919, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115, Rules used = {493, 424, 419} \[ \frac{1}{3} \sqrt{2} E\left (\sin ^{-1}(x)|-\frac{3}{2}\right )-\frac{1}{3} \sqrt{2} F\left (\sin ^{-1}(x)|-\frac{3}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 493
Rule 424
Rule 419
Rubi steps
\begin{align*} \int \frac{x^2}{\sqrt{1-x^2} \sqrt{2+3 x^2}} \, dx &=\frac{1}{3} \int \frac{\sqrt{2+3 x^2}}{\sqrt{1-x^2}} \, dx-\frac{2}{3} \int \frac{1}{\sqrt{1-x^2} \sqrt{2+3 x^2}} \, dx\\ &=\frac{1}{3} \sqrt{2} E\left (\sin ^{-1}(x)|-\frac{3}{2}\right )-\frac{1}{3} \sqrt{2} F\left (\sin ^{-1}(x)|-\frac{3}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0305914, size = 24, normalized size = 0.77 \[ \frac{1}{3} \sqrt{2} \left (E\left (\sin ^{-1}(x)|-\frac{3}{2}\right )-\text{EllipticF}\left (\sin ^{-1}(x),-\frac{3}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.018, size = 25, normalized size = 0.8 \begin{align*} -{\frac{ \left ({\it EllipticF} \left ( x,{\frac{i}{2}}\sqrt{6} \right ) -{\it EllipticE} \left ( x,{\frac{i}{2}}\sqrt{6} \right ) \right ) \sqrt{2}}{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{-x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{3 \, x^{2} + 2} \sqrt{-x^{2} + 1} x^{2}}{3 \, x^{4} - x^{2} - 2}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{- \left (x - 1\right ) \left (x + 1\right )} \sqrt{3 x^{2} + 2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{2}}{\sqrt{3 \, x^{2} + 2} \sqrt{-x^{2} + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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